Calculate the size of the quantum involved in the excitation of
(a) An electronic oscillation of period 2.50 fs,
(b) A molecular vibration of period 2.21 fs,
(c) A balance wheel of period 1.0 ms. Express the results in joules and kilojoules per mole.

In an X-ray photoelectron experiment, a photon of wavelength 121 pm ejects an electron from the inner shell of an atom and it emerges with a speed of 56.9 Mm S-I. Calculate the binding energy of the electron.

The Planck distribution gives the energy in the wavelength range dA at the wavelength A. Calculate the energy density in the range 650 nm to 655 nm inside a cavity of volume 100 em3 when its temperature is
(a) 25°C,
(b) 3000°C

The Einstein frequency is often expressed in terms of an equivalent temperature θE, where θE= hv/k, Confirm that θE has the dimensions of temperature, and express the criterion for the validity of the high-temperature form of the Einstein equation in terms of it. Evaluate θE for
(a) Diamond, for which V= 46.5 THz and
(b) For copper, for which V= 7.15 THz. What fraction of the Dulong and Petit value of the heat capacity does each substance reach at 25°C?

The ground-state wave function of a hydrogen atom is Ψ = (1/xa3)1/2e-r1ao where ao = 53 pm (the Bohr radius).
(a) Calculate the probability that the electron will be found somewhere within a small sphere of radius 1.0 pm centered on the nucleus.
(b) Now suppose that the same sphere is located at r = av. What is the probability that the electron is inside it?

Normalize the following wave functions:
(a) Sin (nπx/L) in the range 0
(b) A constant in the range -L
(c) e-x/a in three-dimensional space,
(d) xe r/12a in three-dimensional space. Hint: The volume element in three dimensions is dτ = r2dr sin θ dθ dθ, with 0
Use the integral in Example 8.4.

Use mathematical software to construct superposition’s of cosine functions and determine the probability that a given momentum will be observed. If you plot the superposition (which you should), set x = 0 at the centre of the screen and build the superposition there. Evaluate the root mean square location of the packet, (x2) l/2.

(a) Given that any operators used to represent observables must satisfy the commutation relation in eqn 8.38, what would be the operator for position if the choice had been made to represent linear momentum parallel to the x-axis by multiplication by the linear momentum? These different choices are all valid 'representations' of quantum mechanics.

(b) With the identification of x in this representation, what would be the operator for l/x?

We saw in Impact I8.1 that electron microscopes can obtain images with several hundredfold higher resolution than optical microscopes because of the short wavelength obtainable from a beam of electrons. For electrons moving at speeds close to c, the speed of light, the expression for the de Broglie wavelength (eqn 8.12) needs to be corrected for relativistic effects:

Where c is the speed of light in vacuum and V is the potential difference through which the electrons are accelerated.

(a) Use the expression above to calculate the de Broglie wavelength of electrons accelerated through 50 kV.

A star too small and cold to shine has been found by S. Kulkarni, K. Matthews, B.R. Oppenheimer, and T. Nakajima (Science 270, 1478 (1995)). The spectrum of the object shows the presence of methane, which, according to the authors, would not exist at temperatures much above 1000 K. The mass of the star, as determined from its gravitational effect on a companion star, is roughly 20 times the mass of Jupiter. The star is considered to be a brown dwarf, the coolest ever found.
(a) From available thermodynamic data, test the stability of methane at temperatures above 1000 K.
(b) What is Amax for this star?
(c) What is the energy density of the star relative to that of the Sun (6000 K)?
(d) To determine whether the star will shine, estimate the fraction of the energy density of the star in the visible region of the spectrum.

A nitrogen molecule is confined in a cubic box of volume 1.00 m3. Assuming that the molecule has an energy equal to μkT at T = 300 K, what is the value of n = (n~ + nJ + 11 ;) 112for this molecule? What is the energy separation between the levels n and 11+ I? What is its de Broglie wavelength? Would it be appropriate to describe this particle as behaving classically?

Calculate the wavelength of a photon needed to excite a transition between neighbouring energy levels of a harmonic oscillator of effective mass equal to that of an oxygen atom (15.9949 u) and force constant 544 N m-1

Confirm that the wave function for the first excited state of a one-dimensional linear harmonic oscillator given in Table 9.1 is a solution of the Schrödinger equation for the oscillator and that its energy is 3/2 hw).

Assuming that the vibrations of a 14N2molecule are equivalent to those of a harmonic oscillator with a force constant k = 2293.8 N m3, what is the zero-point energy of vibration of this molecule? The mass of a 14Natom is 14.0031 u.

Calculate the separation between the two lowest levels for an 02 molecule in a one-dimensional container of length 5.0 cm. At what value of n does the energy of the molecule reach -tkT at 300 K, and what is the separation of this level from the one immediately below?

The rotation of an lH127I molecule can be pictured as the orbital motion of an H atom at a distance 160 pm from a stationary I atom. (This picture is quite good; to be precise, both atoms rotate around their common centre of mass, which is very close to the I nucleus.) Suppose that the molecule rotates only in a plane. Calculate the energy needed to excite the molecule into rotation. What, apart from 0, is the minimum angular momentum of the molecule?

A small step in the potential energy is introduced into the one-dimensional square-well problem as in Fig. 9.45.
(a) Write a general expression for the first -order correction to the ground-state energy, Ebl).
(b) Evaluate the energy correction for a = L/1O (so the blip in the potential occupies the central 10 per cent of the well), with n = 1.

Calculate the second-order correction to the energy for the system described in Problem 9.6 and calculate the ground-state wave function. Account for the shape of the distortion caused by the perturbation. Hint. The following integrals are useful

The wave function inside a long barrier of height V is Ψ = Ne-nx calculates
(a) The probability that the particle is inside the barrier and
(b) The average penetration depth of the particle into the barrier.

Determine the values of ∂x = ((x2) - (x)2)1/2 and ∂p = ((p2) – (p)2)1/2 for
(a) A particle in a box of length L and
(b) A harmonic oscillator. Discuss these quantities with reference to the uncertainty principle.

The potential energy of the rotation of one CH3 group relative to its neighbour in ethane can be expressed as V( rp)= Vo cos 3rp. Show that for small displacements the motion of the group is harmonic and calculate the energy of excitation from v = 0 to v = 1. What do you expect to happen to the energy levels and wave functions as the excitation increases?

Confirm that the spherical harmonics
(a) Yo,o'
(b) Y2-1 and
(c) Y3+3 satisfy the Schrödinger equation for a particle free to rotate in three dimensions, and find its energy and angular momentum in each case.

Derive an expression in terms of land ml for the half-angle of the apex of the cone used to represent an angular momentum according to the vector model.
Evaluate the expression for an ex spin. Show that the minimum possible angle approaches 0 as 1--7 =.

Derive (in Cartesian coordinates) the quantum mechanical operators for the three components of angular momentum starting from the classical definition of angular momentum, l = r x p. Show that any two of the components do not mutually commute, and find their commutators.

When B3-carotene is oxidized in vivo, it breaks in half and forms two molecules of retinal (vitamin A), which is a precursor to the pigment in the retina responsible for vision (Impact I14,J). The conjugated system of retinal consists of 11 C atoms and one °atom. In the ground state of retinal, each level up to n = 6 is occupied by two electrons. Assuming an average inter nuclear distance of 140 pm, calculate
(a) The separation in energy between the ground state and the first excited state in which one electron occupies the state with n = 7, and
(b) The frequency of the radiation required to produce a transition between these two states.
(c) Using your results and Illustration 9.1, choose among the words in parentheses to generate a rule for the prediction of frequency shifts in the absorption spectra of linear polyenes:
The absorption spectrum of a linear polyene shifts to (higher/lower) frequency as the number of conjugated atoms (increases/decreases).

Carbon monoxide binds strongly to the Fe2+ ion of the haem group of the protein myoglobin. Estimate the vibrational frequency of CO bound to myoglobin by using the data in Problem 9.2 and by making the following assumptions: the atom that binds to the haem group is immobilized, the protein is infinitely more massive than either the C or °atom, the C atom binds to the Fe2+ ion, and binding of CO to the protein does not alter the force constant of the c=o bond.

The particle on a ring is a useful model for the motion of electrons around the porphine ring (2), the conjugated macro cycle that forms the structural basis of the haem group and the chlorophylls. We may treat the group as a circular ring of radius 440 pm, with 22 electrons in the conjugated system moving along the perimeter of the ring. As in Illustration 9.1, we assume that in the ground state of the molecule each state is occupied by two electrons.(a) Calculate the energy and angular momentum of an electron in the highest occupied level.(b) Calculate the frequency of radiation that can induce a transition between the highest occupied and lowest unoccupied levels.

The forces measured by AFM arise primarily from interactions between electrons of the stylus and on the surface. To get an idea of the magnitudes of these forces, calculate the force acting between two electrons separated by 2.0 nm Hints. The Coulombic potential energy of a charge q1 at a distance r from another charge q, is
V = q1q2/4πε 0r
Where ε0 = 8.854 X 10-12 C2 J-1 m-1 is the vacuum permittivity. To calculate the force between the electrons, note that F = -d V/ dr,

We remarked in Impact 19.2 that the particle in a sphere is a reasonable starting point for the discussion of the electronic properties of spherical metal Nan particles. Here, we justify eqn 9.54, which shows that the energy of an electron in a sphere is quantized.

(a) The Hamiltonian for a particle free to move inside a sphere of radius R is

H = - h / 2m ∆2 Show that the Schrödinger equation is separable into radial and angular components. That is, begin by writing Ψ(r, θ, Φ) = X(r) Y (θ, Φ), where X(r) depends only on the distance of the particle away from the centre of the sphere, and Y (θ, Φ) is a spherical harmonic. Then show that the Schrödinger equation can be separated into two equations, one for X, the radial equation, and the other for Y, the angular equation:

You may wish to consult further information 10.1 for additional help.

(b) Consider the case 1= 0. Show by differentiation that the solution of the radial equation has the form

X (r) = (2πR) -1/2 sin (nπr/R)/r

(c) Now go on to show that the allowed energies are given by:

En = n2h2 / 8mR2

This result for the energy (which is eqn 9.54 after substituting m, for m) also applies when l ≠ O.

The Humphreys series is a group of lines in the spectrum of atomic hydrogen. It begins at 12 368 nm and has been traced to 3281.4 nm. What are the transitions involved? What are the wavelengths of the intermediate transitions?

The Li2+ion is hydrogenic and has a Lyman series at 740 747 cm3, 877 924 cm-1, 925 933 cm-1, and beyond. Show that the energy levels are of the form -hcRln2 and find the value of R for this ion. Go on to predict the wave numbers of the two longest-wavelength transitions of the Balmer series of the ion and find the ionization energy of the ion.

W.P. Wijesundera, S.H. Vosko, and F.A. Parpia (Phys. Rev. A 51, 278 (1995)) attempted to determine the electron configuration of the ground state of lawrencium, element 103. The two contending configurations are [Rn] 5f l47s27pl and [Rn] 5f l46d7s2. Write down the term symbols for each of these configurations, and identify the lowest level within each configuration. Which level would be lowest according to a simple estimate of spin-orbit coupling?

Calculate the mass of the deuteron given that the first line in the Lyman series of H lies at 82259.098 cm-1 whereas that of D lies at 82 281.476 cm-1 Calculate the ratio of the ionization energies of H and D

The Zeeman effect is the modification of an atomic spectrum by the application of a strong magnetic field. It arises from the interaction between applied magnetic fields and the magnetic moments due to orbital and spin angular moment a (recall the evidence provided for electron spin by the Stern-Gerlach experiment, Section 9.8). To gain some appreciation for the so-called normal Zeeman effect, which is observed in transitions involving singlet states, consider a p electron, with l= 1 and ml = 0, ±1. In the absence of a magnetic field, these three states are degenerate. When a field of magnitude B is present, the degeneracy is removed and it is observed that the state with ml = +1 moves up in energy by μBB, the state with mj = 0 is unchanged, and the state with m =-1 moves down in energy by μBB, where μB= eh/2m; = 9.274 x 10-24 J T-I is the Bohr magneton (see Section 15.1). Therefore, a transition between a I S O term and a IP 1 term consists of three spectral lines in the presence of a magnetic field where, in the absence of the magnetic field, there is only one.
(a) Calculate the splitting in reciprocal centimeters between the three spectral lines of a transition between a I So term and alp 1 term in the presence of a magnetic field of2 T (where 1 T = 1 kg S-2A-I).
(b) Compare the value you calculated in (a) with typical optical transition wave numbers, such as those for the Balmer series of the H atom. Is the line splitting caused by the normal Zeeman effect relatively small or relatively large?

Explicit expressions for hydrogenic orbitals are given in Tables 10.1 and 9.3.
(a) Verify both that the 3px orbital is normalized (to I) and that 3px and 3dxy are mutually orthogonal.
(b) Determine the positions of both the radial nodes and nodal planes of the 35, 3px' and 3dxy orbitals.
(c) Determine the mean radius of the 35 orbital.
(d) Draw a graph of the radial distribution function for the three orbitals(of part
(b) And discuss the significance of the graphs for interpreting the properties of many-electron atoms.
(e) Create both xy-plane polar plots and boundary surface plots for these orbitals. Construct the boundary plots so that the distance from the origin to the surface is the absolute value of the angular part of the wave function. Compare the 5, p, and d boundary surface plots with that of an [-orbital; e.g. Ψ= x (5z2 - r2) cc sin θ (5 cos2 θ-l) cos Φ.

Some atomic properties depend on the average value of 1/r rather than the average value of r itself. Evaluate the expectation value of 1/, for?
(a) A hydrogen 15 orbital,
(b) A hydrogenic 25 orbital,
(c) A hydrogenic 2p orbital.

The Bohr model of the atom is specified in Problem 10.18. What features of it are untenable according to quantum mechanics? How does the Bohr ground state differ from the actual ground state? Is there an experimental distinction between the Bohr and quantum mechanical models of the ground state?

The wave function of a many-electron closed-shell atom can expressed as a Slater determinant (Section 10Ab). A useful property of determinants is that interchanging any two rows or columns changes their sign and therefore, if any two rows or columns are identical, then the determinant vanishes. Use this property to show that
(a) The wave function is ant symmetric under particle exchange
(b) No two electrons can occupy the same orbital with the same spin.

The distribution of isotopes of an element may yield clues about the nuclear reactions that occur in the interior of a star. Show that it is possible to use spectroscopy to confirm the presence of both 4He+ and 3He+ in a star by calculating the wave numbers of the 11= 3 -7 11= 2 and of the n = 2 -7 n = 1 transitions for each isotope.

The spectrum of a star is used to measure its radial velocity with respect to the Sun, the component of the star's velocity vector that is parallel to a vector connecting the star's centre to the centre of the Sun. The measurement relies on the Doppler effect in which radiation is shifted in frequency when the source is moving towards or away from the observer. When a star emitting electromagnetic radiation of frequency v moves with a speed s relative to an observer, the observer detects radiation of frequency v receding= v f or v approaching = v/f, where f= {(1 – s/c)/(l + s/c)}1/2 and c is the speed of light. It is easy to see that v receding < v and a receding star is characterized by a red shift of its spectrum with respect to the spectrum of an identical, but stationary source. Furthermore, v approaching > v and an approaching star is characterized by a blue shift of its spectrum with respect to the spectrum of an identical, but stationary source. In a typical experiment, v is the frequency of a spectral line of an element measured in a stationary Earth-bound laboratory from a calibration source, such as an arc lamp. Measurement of the same spectral line in a star gives v star and the speed of recession or approach may be calculated from the value of v and the equations above.

(a) Three Fe I lines of the star HDE 271 182, which belongs to the Large Magellanic Cloud, occur at 438.882 nm, 441.000 nm, and 442.020 nm. The same lines occur at 438.392 nm, 440.510 nm, and 441.510 nm in the spectrum of an Earth-bound iron arc. Determine whether HDE 271 182 is receding from or approaching the Earth and estimate the star's radial speed with respect to the Earth.

(b) What additional information would you need to calculate the radial velocity of HDE 271 182 with respect to the Sun?